Hypothesis tests

Appendices

Chi-Square Test for Independence

This lesson explains how to conduct a
chi-square test for independence.
The test is applied when you have two
categorical variables
from a single population. It is used to determine whether
there is a significant association between the two variables.

For example, in an election survey, voters might be classified
by gender (male or female) and voting preference (Democrat,
Republican, or Independent). We could use a chi-square test for
independence to determine whether gender is related to
voting preference. The
sample problem at the end of the lesson
considers this example.

When to Use Chi-Square Test for Independence

The test procedure described in this lesson is appropriate when the
following conditions are met:

If sample data are displayed in a
contingency table,
the expected frequency count for each cell of the table is
at least 5.

This approach consists of four steps: (1) state the hypotheses,
(2) formulate an analysis plan, (3) analyze sample data, and
(4) interpret results.

State the Hypotheses

Suppose that Variable A has r levels, and
Variable B has c levels.
The
null hypothesis states that knowing the level of
Variable A does not help you predict the level of
Variable B. That is, the variables are independent.

Analyze Sample Data

Using sample data, find the
degrees of freedom, expected frequencies,
test statistic, and the P-value associated with the test statistic.
The approach described in this section is illustrated in the
sample problem at the end of this lesson.

where r is the number of levels for one catagorical variable, and
c is the number of levels for the other categorical
variable.

Expected frequencies. The expected frequency counts
are computed separately for each level of one categorical variable
at each level of the other categorical variable. Compute r * c
expected frequencies, according to the following formula.

Er,c = (nr * nc) / n

where
Er,c is the expected frequency count for
level r of Variable A and
level c of Variable B,
nr is the total number of sample observations at
level r of Variable A,
nc is the total number of sample observations at
level c of Variable B, and
n is the total sample size.

Test statistic. The test statistic is a chi-square random variable
(Χ2) defined by
the following equation.

Χ2 =
Σ [ (Or,c - Er,c)2 / Er,c ]

where
Or,c is the observed frequency count at level r of
Variable A and level c of Variable B, and
Er,c is the expected frequency count at level r of
Variable A and level c of Variable B.

P-value. The P-value is the probability of observing a
sample statistic as extreme as the test statistic. Since the
test statistic is a chi-square, use the
Chi-Square Distribution Calculator
to assess the probability associated with the test statistic. Use
the degrees of freedom computed above.

Interpret Results

If the sample findings are unlikely, given
the null hypothesis, the researcher rejects the null hypothesis.
Typically, this involves comparing the P-value to the
significance level,
and rejecting the null hypothesis when the P-value is less than
the significance level.

Test Your Understanding

Problem

A public opinion poll surveyed a simple random sample of 1000 voters.
Respondents were classified by gender (male or female) and by
voting preference (Republican, Democrat, or Independent).
Results are shown in the
contingency table below.

Voting Preferences

Row total

Rep

Dem

Ind

Male

200

150

50

400

Female

250

300

50

600

Column total

450

450

100

1000

Is there a gender gap? Do the men's voting preferences differ
significantly from the women's preferences? Use a 0.05
level of significance.

Solution

The solution to this problem takes four steps:
(1) state the hypotheses, (2) formulate an analysis plan,
(3) analyze sample data, and (4) interpret results.
We work through those steps below:

State the hypotheses. The first step is to
state the
null hypothesis and an alternative hypothesis.

Analyze sample data. Applying the chi-square
test for independence to sample data, we compute
the degrees of freedom,
the expected frequency counts, and
the chi-square test statistic. Based on the
chi-square statistic and the
degrees of freedom, we determine the
P-value.

where
DF is the degrees of freedom,
r is the number of levels of gender,
c is the number of levels of the voting preference,
nr is the number of observations from level r
of gender,
nc is the number of observations from level c
of voting preference,
n is the number of observations in the sample,
Er,c is the expected frequency count when gender is level
r and voting preference is level c, and
Or,c is the observed frequency count when gender is level
r voting preference is level c.

The P-value is the probability that a chi-square statistic
having 2 degrees of freedom is more extreme than 16.2.

Interpret results. Since the P-value (0.0003) is
less than the significance level (0.05), we cannot accept the
null hypothesis. Thus, we conclude that there is a relationship
between gender and voting preference.

Note: If you use this approach on an exam, you may also want to mention
why this approach is appropriate. Specifically, the approach is
appropriate because the sampling method was simple random sampling, the
variables under study were categorical, and the expected frequency count
was at least 5 in each cell of the contingency table.